Resonances in Bounded Media

Amundsen, David E.

doi:10.1111/sapm.12181

Cited by 3 publications

(9 citation statements)

References 17 publications

(28 reference statements)

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“…With the focus being on the form of the boundary conditions and nonlinearity, we consider a slightly modified version of the general model problem in Ref. having a simplified linear operator but where the nonlinear term is more general. In particular, we consider problems of the form

u_{t t} - u_{x x} + f false(u, u_{x} false) = 0,

where

f (u, u_{x})

is a nonlinear function of

u (x, t)

and

u_{x} false(x, t false)

over the domain

x \in [0, 1]

and

t \in [0, \infty)

.…”

Section: Formulation Of Nonlinear Model Problemsmentioning

confidence: 99%

“…In the case that the linear spectrum is sufficiently incommensurate, the solution is given by

u_{1} false(x, θ false) = A \sin false(ω_{0} x false) \sin false(θ false),

where A is the amplitude. For cases where the eigenvalue spectrum is commensurate or nearly‐commensurate, a multiple‐mode formulation is required …”

Section: Formulation Of Nonlinear Model Problemsmentioning

confidence: 99%

“…We again use a very similar method to the one utilized in the previous case to numerically approximate the solution of Equation . Then using an appropriately truncated multimodal solution, an analytic approximation may be constructed and compared as shown in Figure , and we see that it is effectively single mode. Case 3 :

α = 0

. …”

Section: Formulation Of Nonlinear Model Problemsmentioning

confidence: 99%

“…As introduced in Ref. , we will consider a model form which contains the fundamental elements of variation of the linear spectrum combined with nonlinearity and boundary effects in a simpler context. With an underlying linear characteristic structure, as well as a simplified nonlinear structure, the interplay between the linear spectrum and response can be studied in more detail.…”

Section: Introductionmentioning

confidence: 99%

“…The primary focus in Ref. is on the parametric variation of the underlying linear operator, as would arise under geometric variation in the acoustic context. In this study, we will focus more specifically on the role of nonlinearity and boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Resonances in bounded media: Nonlinear and boundary effects

Shatnawi

Amundsen

2019

Stud Appl Math

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We study the response of nonlinear wave systems in bounded domains at or near resonance. There are typically two qualitatively distinct types of response which may be observed relating to whether or not higher harmonics are themselves resonant. We introduce a variety of nonlinear model problems at or near resonance and study the subsequent response. We explain how the features of this problem such as the form of nonlinearity, boundary conditions, and the nature of spectrum play a fundamental role in the qualitative nature of the response. Numerical simulations are carried out to provide further explanation and comparison with analytic approximations. The results of this study provide a better understanding of the impact and interplay between nonlinear and boundary effects and thus in turn will contribute to providing new insights into various physically motivated problems in acoustics and other settings. K E Y W O R D SAsymptotic methods, Bounded media, Resonance, Weakly nonlinear analysis 192

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u_{t t} - u_{x x} + f false(u, u_{x} false) = 0,

where

f (u, u_{x})

is a nonlinear function of

u (x, t)

and

u_{x} false(x, t false)

over the domain

x \in [0, 1]

and

t \in [0, \infty)

.…”

Section: Formulation Of Nonlinear Model Problemsmentioning

confidence: 99%

“…In the case that the linear spectrum is sufficiently incommensurate, the solution is given by

u_{1} false(x, θ false) = A \sin false(ω_{0} x false) \sin false(θ false),

where A is the amplitude. For cases where the eigenvalue spectrum is commensurate or nearly‐commensurate, a multiple‐mode formulation is required …”

Section: Formulation Of Nonlinear Model Problemsmentioning

confidence: 99%

α = 0

. …”

Section: Formulation Of Nonlinear Model Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Resonances in bounded media: Nonlinear and boundary effects

Shatnawi

Amundsen

2019

Stud Appl Math

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Preface: Special Issues in Memory of David Benney (Part II)

Ablowitz

Yang

2017

Stud Appl Math

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Resonantly Forced Waves and the Effect of Tank Geometry on the Free Surface

Cheng¹

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This thesis examines the evolution of small-amplitude, resonantly forced oscillations of fluid in a shallow tank of finite extent. A rectangular tank and annular tank shape are both considered. The fluid in each tank is subject to periodic lengthwise or radial forcing, respectively. The resonant forcing of fluid in a rectangular tank has been examined in detail in prior work. How varying the geometry of the tank affects the evolution of the signal on the free surface is not yet fully understood. This work aims to contribute toward a fuller understanding of the interplay between geometric and resonant effects in axisymmetric geometries.I would also like to express my profound gratitude to my parents, Nelson Cheng and Felicia Nicolaiciuc. Their love, support, and humorous video calls were invaluable to me during my degree. Further thanks go out to my siblings, Nick, Alina, and Andreea, and my many friends in Windsor, London, Waterloo, Toronto, and Ottawa. I am truly blessed to have you all in my life.My final note of appreciation goes out to my wonderful partner, Ryan Walsh, whose confidence in me never wavered. Thank you.

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Resonances in Bounded Media

Cited by 3 publications

References 17 publications

Resonances in bounded media: Nonlinear and boundary effects

Resonances in bounded media: Nonlinear and boundary effects

Preface: Special Issues in Memory of David Benney (Part II)

Resonantly Forced Waves and the Effect of Tank Geometry on the Free Surface

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